We complete the proof of the Generalized Smale Conjecture, apart from the case of
R
P
3
RP^3
, and give a new proof of Gabai’s theorem for hyperbolic
3
3
-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except
S
3
S^3
and
R
P
3
RP^3
, as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a
3
3
-manifold
X
X
, the inclusion
Isom
(
X
,
g
)
→
Diff
(
X
)
\operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X)
is a homotopy equivalence for any Riemannian metric
g
g
of constant sectional curvature.